Treating only
How can mass exist without matter?
and not the context, it is important to understand the usual relativistic definition of mass as the norm of the energy-momentum four-vector $$ \left( m c^2 \right)^2 = \left|\mathbf{p}\right|^2 = E^2 - \vec{p}^2 c^2\,.$$
In this context any system of photons not all pointing in the same direction has mass without having any "matter" in a conventional understanding.
1. Void vs Vacuum
The first thing that needs to be done is to distinguish between void and space (ie vacuum).
Space is not nothing, because you can move things in it; think of it as the medium in which particles can move.
For if space was exactly nothing; then where could you put a particle? There is no place you can put it.
2. Crumpling space
The second thing is to imagine how you can warp this space; this is difficult using the space we actually live in.
So let's imagine a page torn out of Ryders QFT is our space. This is easy enough to warp - you can roll it into a cylinder, or crumple it in some other way.
3. Geodesics or straight lines
But how to do physics on this surface? Well, let's just take Newton's first law: a particle without any forces acting on it moves in a straight line.
When the page is rolled out flat on the surface of a table, the path this particle takes is easy enough to imagine - it's just the straight line we can draw by eye.
But how about when it's crumpled? Well, to make things easy for us let's imagine that the page has been crumpled and glued into a sphere. So how can we draw a straight line on this sphere? We can't do the obvious thing and just 'drill' through the sphere in the obvious straight line - because the interior of the sphere is not space but void - so a particle can't go there.
And we can't do the second most obvious thing either, which is just draw the straight line by eye on the surface of this sphere, because between any two points it's not clear what is the straight line that connects them - because to our eye they all look curved.
We turn this around, by asking if there is some unique property that characterises the straight line on the surface; well: on the surface of the paper spread out flat on the surface of a table, our original set-up, we see straight-away that the straight line between two points is the shortest path between two points.
4. Newton's first law again
So, we use this property on the sphere; and now it's easy to see how a particle will follow Newton's first law on this 'curved' or 'warped' space; it's still moves in a straight line, but here we call them geodesics.
In fact, this sphere is 'warped' in a way that the crumpled page is not; because if you draw triangles on it - which we can now, since we know what straight lines look like - we discover that their interior angles always add up to more than 180 degrees.
5. And GR, briefly
And this is how GR works: given a mass distribution in a space, this tells what the geodesics are in this space - ie how it's curved; and then particles move along these geodesics.
Best Answer
I'll answer, and don't think it's a bad question. We don't know what dark matter (DM) is but do know a few things about DM that make your proposed cause of it not likely, or not relevant. But you do bring up some possibilities in the early universe, although not exactly like what is thought possible. I explain below.
In any event I encourage to keep asking, and answering, the downvotes and rules are easy to get used to and understand a bit how to deal with. encourage you to read more on cosmology and gravitation, if it interests you.
The main reason your proposed mechanism for DM is not possible or relevant is that is does not explain how DM is different than regular matter. Your explanation holds equally if it was normal matter. First, your proposed early universe high curvature that could have attracted DM, also would have attracted normal matter. It would be no different. And it would have continued to do so and set the nucleus for stars and galaxies. Secondly, as @Mladen said in the comment, dark matter does not interact significantly with regular matter or with itself (it does interact gravitationally which is how we detect it), while regular matter does. This is known because we have astronomical evidence of two galaxies passing through each other with the regular matter slowed down by collisions/interactions, and DM not that much affected (with the dark matter density estimated by the gravitational effects). See the bullet cluster for this at https://en.m.wikipedia.org/wiki/Bullet_Cluster.
So, even if you are right, your explanation would be right for both DM and normal matter. Nothing that explains dark matter.
Now, as for the possibility of high curvature early in the universe, with some areas much higher than other, you have to be careful as to when. We know that after 380000 years the cosmic microwave background (CMB) was let loose (after decoupling), and it is extremely homogeneous and isotropic. We don't see anything that looks like there were were disparate areas, with some really high curvature someplace. So, it must have been way before or way after. After recombination we know and see a lot about the universe evolution, and there's no evidence nor reason for it. Even before we don't see much reasons. See the standard possible thoughts on what DM could be at https://en.m.wikipedia.org/wiki/Dark_matter
The very early universe, before inflation, could possibly have formed very large strings and string walls (from string theory), and have caused topological defects. This could be some of what was around during the Planck time and some may have been leftover as relics, according to string theory, if string theory is true in some way (which remains TBD, with less optimism for it that there used to be). But at those early times if there were string or string walls, gravitation would still not have been what we envision now, curvature, it would have been more stringy things interacting, in ways we don't know. Most thoughts of the DM is that it represents some relic particles formed in the early universe. If formed then, doesn't matter if they would have been attracted by a high curvature.
But we've not been able to identify any dark matter particles, and it is still a research area.
See a summary PDF on some possible early universe relics at https://arxiv.org/abs/1202.5851. There are other related articles, but nothing definitive.
CLARIFICATION EDIT RESPONSES
The clarification to the question is whether some 'strange and strong' warping of spacetime might be what creates the effects we call DM. So she is not asking whether the warping can create exotic new particles that are then the DM, but where it is the DM.
There's some reasons why that's not likely, but first let me say something about would need to be involved to create something like that, in my view. There would have to be vacuum macroscopic solutions of GR where is is some stable region with a gravitational field similar to that created by some distribution of matter locally (since that is what it looks to us, e.g., in the halo of galaxies). The geons that Wheeler proposed seem to not be stable, but not proven. The other possibility involves some other semi-stable, exact solution of GR in vacuum where the gravitational field looks like it's caused in a local region. There are GR vacuum solution with a semi-stable or stable region of high gravity. An example are solitons, which can be in the form of soliton waves or other configurations, and with possible configurations like kinks and walls. Those have been explored and it is hard to find those kinds of solutions. Since GR is nonlinear you can't add one solution to the other one, the effects that cause the solitons are nonlinear. They do not exist in the linear approximations. It is thought that some of those could have been relics from the early universe, as in my reference above. There could be other solutions that allow it that have not been found. See the classic treatment for gravitational solitons at https://www.abebooks.co.uk/Gravitational-Solitons-V-BELINSKI-E-VERDAGUER/18734763538/bd. It's a book by those two authors, and not cheap, and you can't read it online for free. There's been more papers over the years, and you can google gravitational solitons. And there could some other solutions that fit better the DMs we see.
WHY UNLIKELY
First, they would have had to have been created early in the universe with initially a pretty homogeneous and isotropic distributions in the large. Then thaose would have had to have the property that they not be that cohesive, i.e., that a very large number of much smaller lumps can happen, it, they'd have to be splittable into mass particle-like things, or smaller than galaxy lumps of those. That seems highly unlikely unless they were particle or so sized, i.e., each one was a DM particle. And now we're back where we started.
Postulating any other form would require backing via calculations of some of those soliton solutions and how they might break up into smaller ones. I have not seen any. It's just not easy in GR to find general solutions.
So if you wish to try to postulate that, there's a lot of calculations to be done to show plausibility.